Chapter 3 Vector Spaces

The operations of addition and scalar multiplication are used in many contexts in mathematics. Regardless of the context, however, these operations usually obey the same set of algebra rules. Thus a general theory of mathematical systems involving addition and scalar multiplication will have application to many areas in mathematics. Mathematical systems of this form are called vector spaces or linear spaces.

1 Definition and Examples Euclidean Vector Spaces R n In general, scalar multiplication and addition in R n are defined by and for anyand any scalar 。

The Vector Space R m×n R m×n denote the set of all m × n matrices with real entries.

Definition Let V be a set on which the operations of addition and scalar multiplication are defined. By this we mean that, with each pair of elements x and y in V, we can associate a unique elements x+y that is also in V, and with each element x in V and each scalar, we can associate a unique element x in V. The set V together with the operations of addition and scalar multiplication is said to form a vector space if the following axioms are satisfied. Vector Space Axioms

A1. x+y=y+x for any x and y in V. A2. (x+y)+z=x+(y+z) for any x, y, z in V. A3. There exists an element 0 in V such that x+0=x for each x ∈ V. A4. For each x ∈ V, there exists an element –x in V such that x+(-x)=0. A5. α(x+y)= αx+ αy for each scalar α and any x and y in V. A6. (α+β)x=αx+βx for any scalars α and β and any x ∈ V. A7. (αβ)x=α(βx) for any scalars α and β and any x ∈ V. A8. 1·x=x for all x ∈ V.

The closure properties of the two operations: C1. If x ∈ V and α is a scalar, then αx ∈ V. C2. If x,y ∈ V, then x+y ∈ V. Example Let W={(a,1) a real} with addition and scalar Multiplication defined in the usual way.

Example Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on S by We use the symbol to denote the addition operation for this system avoid confusion with the usual addition x+y of row vectors. Show that S, with the ordinary scalar multiplication and addition operation,is not a vector space. Which of the eight axioms fail to hold?

The Vector Space C[a, b] C[a,b] denote the set of all real-valued functions that are defined and continuous on the closed interval [a,b].

The Vector Space P n P n denote the set of all polynomials of degree less than n.

Theorem If V is a vector space anf x is any Element of V, then (1) 0x=0. (2) x+y=0 implies that y=-x. (3) (-1)x=-x.

2 Subspaces Example Let, S is a subset of R 2. Definition If S is a nonempty subset of a vector space V, and S satisfies the following conditions: (1) whenever for any scalar (2) whenever and then S is said to be a subspace of V.

Example 1 、 Let, S is a subspace of R 3. 2 、 Let. If either of the two conditions in the definition fails to hold, then S will not be a subspace. 3 、 Let.The set S forms a subspace of R 2 × 2.

The Nullspace of a Matrix Let A be an m × n matrix. Let N(A) denote the set of all solutions to the homogeneous system Ax=0. Thus The subspace N(A) is called the nullspace of A. Example Determine N(A) if

The Span of a Set of Vectors Definition Let v 1, v 2, …, v n be vectors in a vector space V. A sum of the form α 1 v 1 + α 2 v 2 + ‥‥ α n v n, where α 1, …, α n are scalars, is called a linear combination of v 1, v 2, …, v n. The set of all linear combinations of v 1, v 2, …, v n is called the span of v 1, v 2, …, v n. The span of v 1, v 2, …, v n will be denoted by Span(v 1, …, v n ).

Theorem If v 1, v 2, …, v n are elements of a vector space V, then Span(v 1, v 2, …, v n ) is a subspace of V. Spanning Set for a Vector Space Definition The set {v 1, v 2, …, v n } is a spanning set for V if and only if every vector in V can be written as a linear combination of v 1, v 2, ‥‥, v n.

Example Which of the following are spanning sets for R 3 ?

3 Linear Independence Consider the following vectors in R 3 :

Conclusion: (1) If v 1, v 2, …, v n span a vector space V and one of these vectors can be written as a linear combination of the other n-1 vectors, then those n-1 vectors span V. (2) Given n vectors v 1, v 2, …, v n, it is possible to write one of the vectors as a linear combination of the other n-1 vectors if and only if there exist scalars c 1, …, c n not all zero such that c 1 v 1 +c 2 v 2 + ‥‥ c n v n =0

Definition The vectors v 1, v 2, …, v n in a vector space V are said to be linearly independent if c 1 v 1 +c 2 v 2 + ‥‥ +c n v n =0 implies that all the scalars c 1, …, c n must equal 0.

Definition The vectors v 1, v 2, …, v n in a vector space V are said to be linearly dependent if there exist scalars c 1, …, c n not all zero such that c 1 v 1 +c 2 v 2 + ‥‥ +c n v n =0 If there are nontrivial choices of scalars for which the linear combination c 1 v 1 +c 2 v 2 + ‥‥ +c n v n equals the zero vector, then v 1, v 2, …, v n are linearly dependent. If the only way the linear combination c 1 v 1 +c 2 v 2 + ‥‥ +c n v n can equal the zero vector is for all the scalars c 1, …, c n to be 0, then v 1, v 2, …, v n are linearly independent.

Theorems and Examples Example Which of the following collections of vectors are linearly independent in R 3 ?

Theorem If x 1, x 2, …, x n be n vectors in R n and let X=(x 1, …, x n ). The vectors x 1,x 2, …, x n will be linearly dependent if and only if X is singular. Example Determine whether the vectors (4, 2, 3) T, (2, 3, 1) T, and (2, -5, 3) T are linearly dependent.

Example Given determine if the vectors are linearly independent.

Theorem If v 1, v 2, …, v n be vectors in a vector space V. A vector v in Span(v 1, …, v n ) can be written uniquely as a linear combination of v 1,v 2, …, v n if and only if v 1,v 2, …, v n are linearly independent.

4 Basis and Dimension Definition The vectors v 1, v 2, …, v n form a basis for a vector space V if and only if (1) v 1, …, v n are linearly independent. (2) v 1, …, v n span V.

Example The standard basis for R 3 is {e 1, e 2, e 3 };however, there are many bases that we could choose for R 3. Example In R 2×2, consider the set {E 11, E 12, E 21, E 22 }, where

Theorem If {v 1, v 2, …, v n } is a spanning set for a vector space V, then any collection of m vectors in V, where m>n, is linearly dependent. Corollary If {v 1, v 2, …, v n } and {u 1, u 2, …, u m } are both bases for a vector space V, then n=m.

Definition Let V be a vector space. If V has a basis consisting of n vectors, we say that V has dimension n. The subspace {0} of V is said to have dimension 0. V is said to be finite- dimensional if there is a finite set of vectors that spans V; otherwise, we say that V is infinite-dimensional.

Theorem If V is a vector space of dimension n>0: (1) Any set of n linearly independent vectors spans V. (2) Any n vectors that span V are linearly independent. Example Show that is a basis for R 3.

Theorem If V is a vector space of dimension n>0, then: (1) No set of less than n vectors can span V. (2) Any subset of less than n linearly independent vectors can be extended to form a basis for V. (3) Any spanning set containing more than n vectors can be pared down to form a basis for V.

5 Change of Basis Changing Coordinates in R 2 x=x1e1+x2e2x=x1e1+x2e2 the coordinate of x is (x 1, x 2 ) T x=αy+βzx=αy+βzthe coordinate of x is (α, β) T

Changing Coordinates Let [e 1, e 2 ] be the standard basis, [u 1, u 2 ] is another basis. Two problems: (1) Given a vector x=(x 1, x 2 ) T, find its coordinates with respect to u 1 and u 2. (2) Given a vector c 1 u 1 +c 2 u 2, find its coordinates with respect to e 1 and e 2.

x=Uc the matrix U is called the transition matrix from the ordered basis [u 1, u 2 ] to the basis [e 1, e 2 ]. Example Let u 1 =(3,2) T, u 2 =(1,1) T, and x=(7,4) T. Find the coordinates of x with respect to u 1 and u 2.

Example Let b 1 =(1,-1) T, b 2 =(-2,3) T. Find the transition matrix from [e 1, e 2 ] to [b 1, b 2 ] and determine the coordinates of x=(1,2) T with respect to [b 1, b 2 ]. Example Find the transition matrix corresponding to the change of basis from [v 1, v 2 ] to [u 1, u 2 ], where and

Change of Basis for a General Vector Space Definition Let V be a vector space and let E=[v 1, v 2, …, v n ] be an ordered basis for V. If v is any element of V, then v can be written in the form v=c 1 v 1 +c 2 v 2 + ‥‥ +c n v n where c 1, …, c n are scalars. Thus we can associate with each vector v a unique vector c=(c 1, c 2, …, c n ) T in R n. The vector c defined in this way is called the coordinate vector of v with respect to the ordered basis E and is denoted [v] E. The c i ’s are called the coordinates of v relative to E.

Example Let E=[v 1, v 2, v 3 ]=[(1, 1, 1) T, (2, 3, 2) T, (1, 5, 4) T ] F=[u 1, u 2, u 3 ]=[(1, 1, 0) T, (1, 2, 0) T, (1, 2, 1) T ] Find the transition matrix from E to F. If x=3v 1 +2v 2 -v 3 and y=v 1 -3v 2 +2v 3 find the coordinates of x and y with respect to the ordered basis F.

6 Row Space and Column Space Definition If A is an m × n matrix, the subspace of R 1 × n spanned by the row vectors of A is called the row space of A. The subspace of R m spanned by the column vectors of A is called the column space of A. Theorem Two row equivalent matrices have the same row space.

Definition The rank of a matrix A is the dimension of the row space of A. Example Let Theorem (Consistency Theorem for Linear Systems) A linear system Ax=b is consistent if and only if b is in the column space of A.

Theorem Let A be an m × n matrix. The linear system Ax=b is consistent for every b ∈ R m if and only if the column vectors of A span R m. The system Ax=b has at most one solution for every b ∈ R m if and only if the column vectors of A are linearly independent. Corollary An n × n matrix A is nonsingular if and only if the column vectors of A form a basis for R n.

Definition The dimension of the nullspace of a matrix is called the nullity of the matrix.

Theorem ( The Rank-Nullity Theorem) If A is an m × n matrix, then the rank of A plus the nullity of A equals n. Example Let Find a basis for the row space of A and a basis for N(A). Verify that dim N(A)=n-r.

Theorem If A is an m × n matrix, the dimension of the row space of A equals the dimension of the column space of A. Example Let Find a basis for the column space of A.

Example Find the dimension of the subspace of R 4 spanned by